# Series- related word problem

1. Jul 29, 2007

### rhymes116

1. The problem statement, all variables and given/known data

Here is the word problem:

A cup of coffee contains about 120 mg of caffeine, while the half life of the caffeine in an adult is about 3.5 hours.

1.) Suppose you ingest a cup of coffee every 6 hours. Obtain an expression for the amount of caffeine , A sub n, remaining in your body just before you drink your nth cup of coffee. Determine limit of [A sub n] as n approaches infinite.

2.) Suppose you require a minimum of 70mg of caffeine to enjoy its benefits. . What is the longest time you can wait between successive cups pf coffee? (in hrs and minutes.)

2. Relevant equations

I believe the only relevant equation for this problem would be taking the half life of caffeine every 3.5 hours

First assuming that a person doesn't take any more caffeine after the first cup, we have the following amounts of caffeine every 3.5 hours:

end of 3.5 hrs=60mg
7hrs=30mg
10.5hrs=15mg
14hrs=7.5mg
17.5hrs=3.75mg
21hrs=1.875mg
24.5hrs=0.9375mg
28hrs=0.46875mg

we see the amount of caffeine approaching zero, correct?

now I am having trouble correlating the amount of caffeine remaining after 6 hours. After 7 hrs we see that 2 half lives have occured. So if I can find out how much remains after 6 hours, then add the additional cup a series can be created. This is where I need help. Anyone have any idea?

3. The attempt at a solution

see above